So I'm having a hard time performing a 3-string sweep, which looks like this;


And you just play this thing over and over again pretty fast.

I can perform the following sort of sweeps fast without any problems though;


So what I'm wondering about is pretty much how I should move my picking-hand, or at least tips how you guys do it, with up/down-strokes since I'm having problems jumping from G to e so darn fast.
you could try it without sweeping at all: V-pulloff-^-V-V

so both the G and e strings during the jump will be hit with downstrokes.
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Yeah I do pull-offs, I can do the same "sweep" mirrored, I can do the 3-string sweeps as I said in the post, and I can do plenty different 5-string sweeps, just having problems with that particular sweep D:
The typical way I was told to do 3 string sweeps was always start the low string with a down stroke. It's typical sweeping motion.

Though after you get the idea of sweeping and how it's done you should be able to come up with your own picking variations that are more comfortable for you
How slow or fast are you practicing?

Are you getting extra string noise?

Is your left hand missing the frets?